DP color functions versus chromatic polynomials for hypergraphs (I)

For a hypergraph $\mathcal{H}$, the DP color function $P_{DP}(\mathcal{H},k)$ of $\mathcal{H}$ is an extension of the chromatic polynomial $P(\mathcal{H},k)$ with the property that $P_{DP}(\mathcal{H},k) \le P(\mathcal{H},k)$ for all positive integers $k$. In this article, we primarily investigate the influence of the minimum cycle length on the DP-coloring function, as well as the relevant properties of the DP-coloring function of $\mathcal{H} \vee K_p$ (i.e., the join of $\mathcal{H}$ and $K_p$). We show that for any linear and uniform hypergraph $\mathcal H$ with even girth, there exists a positive integer $N$ such that $P_{DP} (\mathcal H, k) < P(\mathcal H, k)$ for all integers $k\ge N$, and this conclusion also holds for any hypergraph $\mathcal{H}$ that contains an edge $e$ with the properties that $\mathcal{H}-e$ has exactly $|e|-1$ components and any shortest cycle in $\mathcal{H}$ containing $e$ is an even cycle. For the hypergraph $\mathcal{H}\vee K_p$, we prove that if $\mathcal{H}$ is uniform, then there exist positive integers $p$ and $N$ such that $P_{DP}(\mathcal{H} \vee K_p,k)=P(\mathcal{H} \vee K_p,k)$ holds for all integers $k\geq N$.

💡 Research Summary

This paper investigates the relationship between the DP‑color function P₍DP₎(ℋ,k) and the classical chromatic polynomial P(ℋ,k) for finite, simple hypergraphs ℋ. The DP‑color function, introduced as a generalization of list‑coloring, always satisfies P₍DP₎(ℋ,k) ≤ P(ℋ,k) for every positive integer k. The authors focus on three central questions: (1) whether an even girth forces a strict inequality for large k, (2) whether the existence of an edge whose shortest containing cycle has even length yields a strict inequality, and (3) whether joining a hypergraph with a sufficiently large complete graph Kₚ can make the two functions coincide for all sufficiently large k.

The paper begins with a concise review of hypergraph terminology, the definition of proper k‑colorings, and the classical chromatic polynomial. Lemma 1 provides a deletion‑contraction expansion for P(ℋ,k), while Lemma 2 gives the standard deletion‑contraction recurrence. The DP‑color function is defined via k‑fold covers: a family F of partial maps from the vertex set to {1,…,k} such that each edge e has at most k maps with domain e, and a coloring is a total map avoiding all partial maps in F. The DP‑color function P₍DP₎(ℋ,k) is the minimum number of F‑colorings over all possible k‑fold covers F. Lemma 3 supplies a general upper bound for P₍DP₎(ℋ,k) in terms of the number of vertices n, the uniformity r, and the number of edges m. Lemma 4 gives a technical reduction when an edge e has the property that removing e creates exactly |e|−1 components.

Even girth.
For a connected, linear, r‑uniform hypergraph ℋ with even girth z, Lemma 5 expresses the chromatic polynomial as a sum of three parts: a “low‑degree” polynomial f(k), a term (−1)ᶻ t k^{n−z(r−1)+1} coming from the cycles of length z, and a binomial sum over subsets of edges. Because z is even, the sign of the dominant cycle term is positive. Theorem 6 combines this expansion with the general DP‑upper bound from Lemma 3. For sufficiently large k, the positive dominant term outgrows the DP‑upper bound, guaranteeing P₍DP₎(ℋ,k) < P(ℋ,k). Thus any linear, uniform hypergraph with even girth exhibits a strict gap for large k.

Even shortest cycle through an edge.
Theorem 7 provides a sufficient condition based on a single edge e. If c(ℋ−e)=|e|−1 (i.e., removing e splits the hypergraph into exactly |e|−1 components) and the inequality
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